Unramified abelian covers with many points

TL;DR

This paper constructs algebraic curves over finite fields with specific sizes that have record numbers of rational points by using unramified abelian covers, advancing the understanding of maximal point counts.

Contribution

It introduces new methods to produce curves with many points over finite fields using unramified abelian covers, achieving record counts.

Findings

Curves over finite fields with record numbers of points

Unramified abelian covers as a construction tool

Record point counts over fields with 4, 9, 16, and 25 elements

Abstract

We produce curves with a record number of points over the finite fields with $4$, $9$, $16$ and $25$ elements, as unramified abelian covers of algebraic curves.